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# Topic: Hyperelliptic function

###### In the News (Mon 19 Aug 19)

 Hyperelliptic curve - Wikipedia, the free encyclopedia A hyperelliptic function is a function from the function field of such a curve; or possibly on the Jacobian variety on the curve, these being two concepts that are the same for the elliptic function case, but different in this case. Hyperelliptic curves can be used in hyperelliptic curve cryptography in cryptosystems based on the discrete logarithm problem. Hyperelliptic functions were first published by Adolph Göpel (1812-1847) in his last paper Abelsche Transcendenten erster Ordnung (Abelian transcendents of first order) (in Journal für reine und angewandte Mathematik, vol. en.wikipedia.org /wiki/Hyperelliptic_curve   (425 words)

 [No title]   (Site not responding. Last check: 2007-10-23) Related functions compute the sequence of global gap numbers of a divisor or the sequence of gap numbers of a divisor at a place of degree 1. Function Fields: The divisor class group of a global function field may be computed using an algorithm and implementation due to Florian Hess. Function Fields: The Galois group and lattice of subfields may be found for a function field using code implemented by Katharina Geissler. www.umich.edu /~gpcc/scs/magma/text76.htm   (2351 words)

 Places   (Site not responding. Last check: 2007-10-23) The group of divisors of the algebraic function field F/k, which is the free abelian group generated by the elements of the set of places of F/k. A monic prime polynomial in k[x] or 1/x or an ideal, corresponding to the place of the coefficient field of the function field of the place P which P lies above (the function field of P must be a finite extension of k(x)). The completion of the algebraic function field F or an order O of such at the place p of F or the function field of O. The place p must have degree 1 when F or O is not global. magma.maths.usyd.edu.au /magma/htmlhelp/text731.htm   (918 words)

 New Page 0   (Site not responding. Last check: 2007-10-23) A hyperelliptic function field is a field of the form k(x,y) where k is a finite field of odd characteristic and y Although generically, the 3-Sylow subgroup of this ideal class group is small (and frequently trivial), it is possible to generate hyperelliptic function fields -- even infinite families of such fields -- whose 3-rank is unusually large. This talk presents several methods for explictly constructing hyperelliptic function fields of high 3-rank, and more generally, high l-rank for any prime l coprime to the characteristic of k. research.microsoft.com /~klauter/Scheidler.htm   (242 words)

 Math 2000 We discuss recent joint work with C. Pomerance and I. Shparlinski on bounds for the frequency of large and small values of the Carmichael function and of the relevance of such bounds for the distribution of the power generator of pseudorandom numbers. quadratic function fields, and we are in the process of exploring certain cubic function field extensions for this purpose. the cardinality of the Jacobian of the corresponding hyperelliptic curve. www.cms.math.ca /Events/summer00/abs/cnt.html   (1576 words)

 Points on Hyperelliptic Curves The hyperelliptic curve is embedded in a weighted projective space, with weights 1, g + 1, and 1, respectively on x, y and z. Given a point P on a hyperelliptic curve C_1, such that C is a base extension of C_1, returns the corresponding point on the hyperelliptic curve C. The curve C can be, e.g., the reduction of C_1 to finite characteristic (i.e. Given a hyperelliptic curve C defined over a finite field, this function computes the zeta function of C. The zeta function is returned as an element of the function field in one variable over the integers. www.math.niu.edu /help/math/magmahelp/text1026.html   (648 words)

 Rudi Weikard (Home Page)   (Site not responding. Last check: 2007-10-23) Abelian functions are inverses of abelian integrals, i.e., integrals of rational functions of x and an algebraic function of x. If this algebraic function is the square root of a polynomial one is looking (in general) at hyperelliptic integrals and hyperelliptic functions. If the degree of the polynomial is 3 or 4 one has elliptic integrals and elliptic functions while for degree 1 or 2 the integrals may be solved in terms of inverses of trigonometric functions. www.math.uab.edu /~rudi   (407 words)

 Trivia - Niels Henrik Abel From Berlin he passed to Freiberg, and here he made his brilliant researches in the theory of functions: elliptic function, hyperelliptic function, and a new class now known as abelian functions being particularly intensely studied. In 1826 Abel moved to Paris, and during a ten month stay he met the leading mathematicians of France, but he was poorly appreciated, as his work was scarcely known, and his modesty restrained him from proclaiming his researchings. Under Abels guidance, the prevailing obscurities of analysis began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made. mywebpage.netscape.com /Abashiri2434/niels-henrik-abel-trivia.html   (521 words)

 Related Structures   (Site not responding. Last check: 2007-10-23) The rational function field k(x) if the function field F is an extension of k(x) and k if F is an extension of k. A hyperelliptic function field in the Weil restriction over GF(q) of the elliptic function field E: y^2 + xy + x^3 + ax^2 + b defined over GF(q^n) where q is a power of 2. Also returns a function which can be used to map a place (not a pole or zero of x) of F into a divisor of the result. www.umich.edu /~gpcc/scs/magma/text687.htm   (368 words)

 PUBLICATIONS   (Site not responding. Last check: 2007-10-23) Optimized baby-step giant-step methods in hyperelliptic function fields. Sharp upper bounds for arithmetics in hyperelliptic function fields. An Algorithm for Determining the Regulator and the Fundamental Unit of a Hyperelliptic Congruence Function Field. www.uwyo.edu /astein/publications.html   (641 words)

 Hyperelliptic curves   (Site not responding. Last check: 2007-10-23) Noticing that an elliptic curve group is precisely the Jacobian of a hyperelliptic curve of genus 1, it is natural to investigate Jacobians of higher genus hyperelliptic curves as a possible replacement (Koblitz (1989)). In algebraic geometry, it is common to define them by abstract properties of their function fields; on the other hand, the cryptographic community usually assumes a more operational point of view and defines them by concrete curve equations. To the best of my knowledge, this is the first attempt to provide an exhaustive classification of hyperelliptic curves and their imaginary or real quadratic models in even characteristic. www.lix.polytechnique.fr /Labo/Andreas.Enge/diss/diss/node5.html   (327 words)

 Advanced examples Finally, the Dedekind zeta function has a pole at s=1 and we need its residue (or, rather, the residue of zeta^ * (s)) which we compute using the class number formula. This is an example as to how to construct the first 20 coefficients of the L-series of an elliptic curve of conductor 11 without knowing anything about either the curve or modular form theory. The resulting L-function does not satisfy the required functional equation, so something unusual does happen at a prime where both E and K have bad reduction, which in this case must be either p=2 or p=3. www.math.lsu.edu /magma/text1348.htm   (1495 words)

 Prof. Andreas Stein Abstract By considering the function fields of irreducible hyperelliptic curves we can investigate the security of hyperelliptic curve cryptosystems with the help of number-theoretic ideas. Hereby, we provide sharp estimates for the divisor class number of a hyperelliptic function field, i.e. Finally, we show how these methods can be extended to any algebraic function field given a way to compute the group operation in the corresponding Jacobian. www.math.uiuc.edu /Bulletin/Abstracts/November/Nov15_99/stein_nov18-99.html   (184 words)

 PlanetMath: a harmonic function on a graph which is bounded below and nonconstant owned by drini Heaviside unit step function (in one-sided limit) owned by matte holomorphic functions of several variables owned by jirka planetmath.org /encyclopedia/H   (1456 words)

 Analytic Jacobians of Hyperelliptic Curves For a hyperelliptic curve there is a natural choice of holomorphic differentials, namely varphi_i = x^(i - 1)dx/y. Given f in C[x] where C is a fixed precision complex field (see Section Fixed Precision Real Numbers), this function returns the analytic Jacobian of the hyperelliptic curve defined by y^2 = f(x). The other functions all rely on the function LinearRelation and work with greater reliability if a high precision is chosen. modular.fas.harvard.edu /docs/magma/htmlhelp/text1223.htm   (3528 words)

 Research Interests -> José D. Edelstein The link between both constructions is summarized in the statement that the prepotential of the former corresponds to the logarithm of the quasiclassical tau function of the latter. We found that the blowup function is a hyperelliptic sigma function and we described an explicit procedure to expand it in terms of the basic observables of the twisted theory up to arbitrary order. The blowup function is given by a hyperelliptic sigma function with singular characteristic. www.math.ist.utl.pt /~jedels/resprop.htm   (1733 words)

 Abstracts The theory is examplified by the cases of hyperelliptic and trigonal curves. Rational analogies of such Abelian functions are described in terms of Schur functions constructed by the Weierstrass gap sequence. This method permits to obtain multiple integral representations for the correlation functions of the XXZ model in an external constant magnetic field in all the regimes. www.amsta.leeds.ac.uk /~vadim/abstracts.html   (4637 words)

 Rovira: Equations of hyperelliptic modular curves We compute, in a unified way, the equations of all hyperelliptic modular curves. The main tool is provided by a class of modular functions introduced by Newman in 1957. The method uses the action of the hyperelliptic involution on the cusps. www-mathdoc.ujf-grenoble.fr /numdam-bin/item?id=AIF_1991__41_4_779_0   (146 words)

 Glossary: abelian or hyperelliptic function   (Site not responding. Last check: 2007-10-23) An abelian or hyperelliptic function is a generalisation of an It is a function of two variables with four periods. it can also be regarded as the inverse function to certain integrals (called abelian or hyperelliptic integrals) of the form www-groups.dcs.st-and.ac.uk /~history/Glossary/abelian_function.html   (50 words)

 Titles and Abstracts of MAGC Talks This lecture gives an introduction to the use of cyclic groups in public key cryptography, generic attacks like baby-step/giant-step or Pollard-Rho and, motivated by the classical DL in the multiplicative group and by elliptic curves, the approach to use "group schemes" like abelian varieties (without precise definitions of course). Some methods to compute their order are mentioned (ref. to Morain's lecture) and then the relation to ideal class groups of orders in the function fields is explained. Explicit examples will be hyperelliptic function fields as generalizations of elliptic fields. www.math.uiuc.edu /~boston/magctitles.html   (764 words)

 METU MATHEMATICS DEPARTMENT Congruence zeta function, the functional equation for the L-functions. Elliptic functions, modular functions, Dedekind eta function, congruences for the coefficients of the modular function j, Rademacher's series for the par-tition function, modular forms with multiplicative coefficients, Kronecker's theorem, general Dirichlet series and Bohr's equivalence theorem. Holomorphic functions, comparison of one and several variables, domains of holomorphy, subharmonicity, pseudoconvexity, invariant metrics, holomorphic maps, Stein and CR-manifolds, integral formulas, equation. www.math.metu.edu.tr /courses/graduate.shtml   (2274 words)

 Fraunhofer ITWM: Cryptography Hyperelliptic curves* are a generalization of elliptic curves. In contrast to the latter, they allow a larger range of parameters to choose from-resulting in a higher level of security. In this research project, scientists of Fraunhofer ITWM developed algorithms to compute subfields* and automorphisms* as well as to perform explicit calculations in endomorphism rings* of hyperelliptic function fields*. www.itwm.fhg.de /mab/competences/Crypto/index_en.php   (192 words)

 Other Decision based Neural Networks   (Site not responding. Last check: 2007-10-23) A RBF discriminant function is a function of the radius betwenn the pattern and a centroid, Elliptic Basis Function The basic RBF version of the DBNN discussed before is based on the asumption that the feature space is uniformly weighted in all directions. The most general form of a second order basis functions is the (skewed) hyperelliptic basis function. www.gc.ssr.upm.es /inves/neural/ann1/supmodel/decbas.htm   (472 words)

 Harold's Bibliography Stark, H.M., On the zeros of Epstein's zeta function, Mathematika, 14, 1967, 47-55. Stark, H.M., The role of modular functions in a class-number problem, J. Number Theory, 1, 1969, 252-260. Stark, Harold M., Multipath zeta functions of graphs, Emerging applications of number theory (Minneapolis, MN, 1996), 601-615, IMA Vol. www.mtholyoke.edu /~aschwart/stark/papers.html   (1114 words)

 [No title] Even if the reader has never seen the phrase genus-2'' before, he knows that the class of genus-2 hyperelliptic curves is contained in the class of hyperelliptic curves. Can you conclude that a random element of X'' is an element of X? No. It is, by definition, a measurable function to X from the probability space \Pr of possible universes. For example, I once bumped into a cryptographer who had completely misunderstood various theorems because---having never seen the definition of the word random''---he incorrectly assumed that a random function from S to T'' was a function from S to T. You can add to these costs by introducing new non-restrictive adjectives. cr.yp.to /bib/devil-name.html   (703 words)

 Template of MITACS Project Website M. Jacobson, A. Menezes, and A. Stein, "Hyperelliptic Curves and Cryptography", High Primes and Misdemeanors - Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications Series, 41 (2004), 255-282. Y. Lee, R. Scheidler and C. Yarrish, "Computation of the Fundamental Units and the Regulator of a Cyclic Cubic Function Field", Experimental Mathematics 12 (2003), 211-225. Stein and E. Teske, Explicit bounds and heuristics on class numbers in hyperelliptic function fields, Mathematics of Computation, 71 (2002), 837-861. www.cacr.math.uwaterloo.ca /mitacs/Publication.htm   (1854 words)

 Brauer Groups of Local Elliptic and Hyperelliptic Curves and Central Division Algebras Over Their Function Fields ...   (Site not responding. Last check: 2007-10-23) We compute the torsion part of the unramified Brauer group of the function field of any elliptic curve defined over a local field of characteristc zero. For the case of a non dyadic ground field we prove that all elements of the 2-torsion part can be represented by quaternion algebras and describe them in an explicit form. We obtain a similar result for hyperelliptic curves with good reduction such that their defining equation y 2 = f(x) satisfies the condition deg f(x) 6j 0(mod 4).... sherry.ifi.unizh.ch /4620.html   (339 words)

 [No title] Our goal here is to introduce a special function $\xi(x, \lambda)$ on $\Bbb R\times\Bbb R$ associated to $H$ which we believe will be a valuable tool in the spectral and inverse spectral theory. For example, we have shown that the $\xi$ function relating to half-line problems on $[0, \infty)$ with different boundary conditions at $0$ determines the potential uniquely a.e. By a Sobolev estimate and using $\Cal H_{1}(A)\subset\Cal H_{1}(- \frac{d^2}{dx^2})$, $F$ is a functional in $\Cal H_{-1}$, so we write $F(f)=\langle\varphi, f\rangle$ with $\varphi(x)=\delta(x-x_{0})$. www.ma.utexas.edu /mp_arc/html/papers/94-80   (2401 words)

 Réunion d'hiver 2004 de la SMC In this talk, we will discuss several explicit constructions of hyperelliptic function fields whose Jacobian have high 3-rank. Some of the methods are analogues of techniques for generating quadratic number fields of high 3-rank, while others are unique to the function field setting. In particular, a method for increasing the 3-rank by extending the field of constants will be emphasized, which may also be used to increase the l-rank. www.cms.math.ca /Events/winter04/res/nt.f   (2378 words)

 Mathematics of Computation Linear recurrences with polynomial coefficients and computation of the Cartier-Manin operator on hyperelliptic curves. Ideal arithmetic and infrastructure in purely cubic function fields. Index calculus attack for hyperelliptic curves of small genus. www.ams.org /mcom/2005-74-252/S0025-5718-05-01758-8/home.html   (397 words)

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